Remember...they claim to be lifelong Democrats.
I don't see their peculiar behaviour as being associated with political allegiance as much as religious organization. WBC seems to be an Independent Fundamental Primitive Baptist church. They are link to three Baptist Confessions of Faith in the doctrinal statement on their website. Fred Phelps, founder of this local church, attended Bob Jones University for two years. I don't know whether another church sponsored him when he planted WBC in 1955. The previous year he was an associate pastor at East Side Baptist Church in Topeka so they may have had a service to found his church. He stopped fellowship with East Side afterward.
One inherent problem for Fundamentalists, Baptist or non-denominational, is the doctrinal view of church structure. They view the church as always local on earth, and are fiercely protective of the independence of the local church. It is their defence against heresy infecting the churches. They can choose, or decline, to fellowship with like-minded churches. There is no governing authority outside the local church. Churches plant other churches which assume all the rights and responsibilities of a New Testament local church. Doctrinally this is how they view Apostolic Succession. That is likely what happened win the case of Fred Phelps and WBC.